Is 1 = 0.9999......

[bleeb]

Originally posted by: dparker
I'm really bored so I think I'll read this thing through.

Good Luck... I've been a contributor since almost the beginning. I started around the 18th post.

 

[silverpig]

Originally posted by: shadow
As I briefly mentioned before, I am not very skiffful at conveying abstract ideas. So I am going to try again to convey my point to you, and hopefully elucidate in such a way that we turn to the same page.

I am arguing that there is an inherent difficulty dealing with infinite numbers. I have gone through a few of the proofs and I was trying to illustrate this difficulty with my own little equation. Ross, I am not trying to ignore your last post, where you say progressing to infinity is not necessary as all that is needed to do a proof for .999r = 1 is simple math. I'll come out right now and say I do not know why or how that statement can be made, I do not know how one can use simple math to deal with an infinite number. I hope to convey to you now why I find that difficult. Please follow me on this one. I know that in arguements it begins to feel like us vs them and that the other side never takes the time to listen to what you are saying. I opened the page to your proof, took one glance at it's length and promptly closed the window. I have just told you I am guilty of being a bad arguer, I have not taken the time to understand your proof. This much is true. But, (there's always a but though ain't there) with my point, there is no need to grab a piece of paper and follow you through your proof. With my point all I have to do is see one line referencing to infinity in a certain way. For example:

x= 0.999r

Apparently to you this is a perfectly fine statement. I gag. It seems so inocuous to be able to deal with an infinitly repeating value, you just throw a bar over the top, or place an 'r' after the end of it, or in ATOT just place some '...' in there. I do not find this statement to be valid in a proof. I find that it is ok to say 5.999r but to assign an infinitly recurring number to a variable is trouble. It looks fine on paper, look at it up there, it's harmless enough, after all we are not changing the value, we are merely setting up a proof. What happens though when someone sits behind a desk grabs a piece of paper and tries to do that assignment 'longhand'? What happens when you plug that into a computer and run it? I can take that proof and enter it into my computer and ask it if it ends up with x=1. I can trick a prodigy human calculator into working the proof 'without' representations for infinity. What happens when we give up the bar over the numbers or the 'r' or the '....'? Try to actually work through that proof with a pen and paper, calcualtor, computer and what happens? The poor slob with the paper dies before he gets past the first variable assignment, the calculator breaks before all the nines for the first statement are entered, and the computer just sits there, running the first assignment forever.

What I was trying to allude in my first post was that the manipulation of infinite values is a conundrum. It is an inherent contradiction: 'manipulation of infinite values'. Infinite values cannot be manipulated or measured by the mere fact that they are infinite. If you have a proof with an infinite value it automatically fails as a proof. Proofs like hypothesis have to be falsifiable. If it is impossible to ever work through a proof because one would get hung up on an infinite value then such proofs are unfalsifiable and must be rejected. (What about pi you ask? For the sake of this arguement, let us not start another: let me just leave you with 0.3333r = 1/3, meaning certain recurring values can be dealt with when in a non-recurring state; and a plea not argue this point in this thread).

In my first post I made a statement which might make more sense now. Because we cannot measure the difference between two numbers (as the difference is infinitly small) those two numbers must be the same. My position, or should I say my opinion is that even though a value might be infinitly small it is still there, unmeasurable as it may be, it is still there.

Ross, if the page you posted deals with how infinite values can be manipulated, and then offers reasons at to why this might be possible then I am grossly out of line here. But on a hunch , I don't think I have made an ass out of myself just yet.

It appears we both enjoy a good beer, so....

/me tosses good can of beer to Ross

You have some good questions Shadow, and instead of answering directly, maybe I can show you something else.

Remember back in grade school when you were first learning to add 234 + 87? The teacher taught you the concept of "place value" where 234 = 2 hundreds + 3 tens + 4 ones etc. Well in order for things to line up nicely on your page and to make sure everyone was adding the right digits together, the teacher taught you to think of 87 as 0 hundreds + 8 tens + 7 ones.

234
087 +
-------
321

I emphasized the 0 hundreds part because that 0 is really there, we just don't bother writing it. In fact, every number is of the form ...00087.000... but we just never write all the zeros because we'd never be able to write the number. When we add 234 + 87 we are actually dealing with two infinite strings of zeros on each number. Dealing with infinite strings of digits isn't anything new; we do it all the time. So how can you add 2 + 2 if you can't actually write either number down properly?


Now this brings about another point. Since 0 really is ...000.000... or also just 0.000... let's do a simple subtraction of 1.000... and 0.9...

1.00000000...
0.99999999... -
------------------
0.00000000...

If x - b = 0, then x = b

 

[Jamestl]

Hi everyone!


I thought I'd say take this chance to say hi to everyone in this thread and to remind all of you to buckle up, drive safe, and to stay away from power lines!


This message is sponsored by the Counsel for the Advancement of Math Illiterates

 

[Dumbledore]

Originally posted by: silverpig

Originally posted by: shadow
As I briefly mentioned before, I am not very skiffful at conveying abstract ideas. So I am going to try again to convey my point to you, and hopefully elucidate in such a way that we turn to the same page.

I am arguing that there is an inherent difficulty dealing with infinite numbers. I have gone through a few of the proofs and I was trying to illustrate this difficulty with my own little equation. Ross, I am not trying to ignore your last post, where you say progressing to infinity is not necessary as all that is needed to do a proof for .999r = 1 is simple math. I'll come out right now and say I do not know why or how that statement can be made, I do not know how one can use simple math to deal with an infinite number. I hope to convey to you now why I find that difficult. Please follow me on this one. I know that in arguements it begins to feel like us vs them and that the other side never takes the time to listen to what you are saying. I opened the page to your proof, took one glance at it's length and promptly closed the window. I have just told you I am guilty of being a bad arguer, I have not taken the time to understand your proof. This much is true. But, (there's always a but though ain't there) with my point, there is no need to grab a piece of paper and follow you through your proof. With my point all I have to do is see one line referencing to infinity in a certain way. For example:

x= 0.999r

Apparently to you this is a perfectly fine statement. I gag. It seems so inocuous to be able to deal with an infinitly repeating value, you just throw a bar over the top, or place an 'r' after the end of it, or in ATOT just place some '...' in there. I do not find this statement to be valid in a proof. I find that it is ok to say 5.999r but to assign an infinitly recurring number to a variable is trouble. It looks fine on paper, look at it up there, it's harmless enough, after all we are not changing the value, we are merely setting up a proof. What happens though when someone sits behind a desk grabs a piece of paper and tries to do that assignment 'longhand'? What happens when you plug that into a computer and run it? I can take that proof and enter it into my computer and ask it if it ends up with x=1. I can trick a prodigy human calculator into working the proof 'without' representations for infinity. What happens when we give up the bar over the numbers or the 'r' or the '....'? Try to actually work through that proof with a pen and paper, calcualtor, computer and what happens? The poor slob with the paper dies before he gets past the first variable assignment, the calculator breaks before all the nines for the first statement are entered, and the computer just sits there, running the first assignment forever.

What I was trying to allude in my first post was that the manipulation of infinite values is a conundrum. It is an inherent contradiction: 'manipulation of infinite values'. Infinite values cannot be manipulated or measured by the mere fact that they are infinite. If you have a proof with an infinite value it automatically fails as a proof. Proofs like hypothesis have to be falsifiable. If it is impossible to ever work through a proof because one would get hung up on an infinite value then such proofs are unfalsifiable and must be rejected. (What about pi you ask? For the sake of this arguement, let us not start another: let me just leave you with 0.3333r = 1/3, meaning certain recurring values can be dealt with when in a non-recurring state; and a plea not argue this point in this thread).

In my first post I made a statement which might make more sense now. Because we cannot measure the difference between two numbers (as the difference is infinitly small) those two numbers must be the same. My position, or should I say my opinion is that even though a value might be infinitly small it is still there, unmeasurable as it may be, it is still there.

Ross, if the page you posted deals with how infinite values can be manipulated, and then offers reasons at to why this might be possible then I am grossly out of line here. But on a hunch , I don't think I have made an ass out of myself just yet.

It appears we both enjoy a good beer, so....

/me tosses good can of beer to Ross

You have some good questions Shadow, and instead of answering directly, maybe I can show you something else.

Remember back in grade school when you were first learning to add 234 + 87? The teacher taught you the concept of "place value" where 234 = 2 hundreds + 3 tens + 4 ones etc. Well in order for things to line up nicely on your page and to make sure everyone was adding the right digits together, the teacher taught you to think of 87 as 0 hundreds + 8 tens + 7 ones.

234
087 +
-------
321

I emphasized the 0 hundreds part because that 0 is really there, we just don't bother writing it. In fact, every number is of the form ...00087.000... but we just never write all the zeros because we'd never be able to write the number. When we add 234 + 87 we are actually dealing with two infinite strings of zeros on each number. Dealing with infinite strings of digits isn't anything new; we do it all the time. So how can you add 2 + 2 if you can't actually write either number down properly?


Now this brings about another point. Since 0 really is ...000.000... or also just 0.000... let's do a simple subtraction of 1.000... and 0.9...

1.00000000...
0.99999999... -
------------------
0.00000000...

If x - b = 0, then x = b

Ummm isn't 1.00000000
0.99999999(-)
-------------------
0.00000001

therefore x does not equal b

 

[silverpig]

The nines in 0.999999.... and the zeros in 1.00000000... go on forever.

 

[AbsolutZero]

I bent my wookie.

 

[MadRat]

Originally posted by: RossGr
Those, like your ideas, were beliefs. They were never proven true. Is this another example of how you cannot seperate belief from fact?

Is there an echo in here? You seem to believe in your own unique ability to comprehend the difference between objective and subjective arguments. The problem is that you want to suppress the meaning of infinity. Infinity is not an objective concept that you can define with any number. Your need to insist that infinity is some fixed value just amazes me.

 

[RossGr]

Originally posted by: shadow
As I briefly mentioned before, I am not very skiffful at conveying abstract ideas. So I am going to try again to convey my point to you, and hopefully elucidate in such a way that we turn to the same page.

I am arguing that there is an inherent difficulty dealing with infinite numbers. I have gone through a few of the proofs and I was trying to illustrate this difficulty with my own little equation. Ross, I am not trying to ignore your last post, where you say progressing to infinity is not necessary as all that is needed to do a proof for .999r = 1 is simple math. I'll come out right now and say I do not know why or how that statement can be made, I do not know how one can use simple math to deal with an infinite number. I hope to convey to you now why I find that difficult. Please follow me on this one. I know that in arguements it begins to feel like us vs them and that the other side never takes the time to listen to what you are saying. I opened the page to your proof, took one glance at it's length and promptly closed the window. I have just told you I am guilty of being a bad arguer, I have not taken the time to understand your proof. This much is true. But, (there's always a but though ain't there) with my point, there is no need to grab a piece of paper and follow you through your proof. With my point all I have to do is see one line referencing to infinity in a certain way. For example:

x= 0.999r

Apparently to you this is a perfectly fine statement. I gag. It seems so inocuous to be able to deal with an infinitly repeating value, you just throw a bar over the top, or place an 'r' after the end of it, or in ATOT just place some '...' in there. I do not find this statement to be valid in a proof. I find that it is ok to say 5.999r but to assign an infinitly recurring number to a variable is trouble. It looks fine on paper, look at it up there, it's harmless enough, after all we are not changing the value, we are merely setting up a proof. What happens though when someone sits behind a desk grabs a piece of paper and tries to do that assignment 'longhand'? What happens when you plug that into a computer and run it? I can take that proof and enter it into my computer and ask it if it ends up with x=1. I can trick a prodigy human calculator into working the proof 'without' representations for infinity. What happens when we give up the bar over the numbers or the 'r' or the '....'? Try to actually work through that proof with a pen and paper, calcualtor, computer and what happens? The poor slob with the paper dies before he gets past the first variable assignment, the calculator breaks before all the nines for the first statement are entered, and the computer just sits there, running the first assignment forever.

What I was trying to allude in my first post was that the manipulation of infinite values is a conundrum. It is an inherent contradiction: 'manipulation of infinite values'. Infinite values cannot be manipulated or measured by the mere fact that they are infinite. If you have a proof with an infinite value it automatically fails as a proof. Proofs like hypothesis have to be falsifiable. If it is impossible to ever work through a proof because one would get hung up on an infinite value then such proofs are unfalsifiable and must be rejected. (What about pi you ask? For the sake of this arguement, let us not start another: let me just leave you with 0.3333r = 1/3, meaning certain recurring values can be dealt with when in a non-recurring state; and a plea not argue this point in this thread).

In my first post I made a statement which might make more sense now. Because we cannot measure the difference between two numbers (as the difference is infinitly small) those two numbers must be the same. My position, or should I say my opinion is that even though a value might be infinitly small it is still there, unmeasurable as it may be, it is still there.

Ross, if the page you posted deals with how infinite values can be manipulated, and then offers reasons at to why this might be possible then I am grossly out of line here. But on a hunch , I don't think I have made an ass out of myself just yet.

It appears we both enjoy a good beer, so....

/me tosses good can of beer to Ross

Thanks for the brew!

As SilverPig said you ask some very good questions. I am sorry that you cannot take the time to plow through my PDF, I think if you did, you would find that I am very careful with how I handle the sum to infinity.

Let me summerize, first I develop a pattern by addind first .1 to .999....

.1+ .999... = .1 + .9 + .0999... = 1.0999....
now add .01
.01 + .999.... = .01 + .99 + .00999... = 1.00999....

I then show that this can be generalized to show that if I add .1^n I get 1.000...bunch of zeros...999.... The point being that no matter how small a number I add to .999... I will get 1 plus a little more the final statement is

.1^n + .999... > 1 where I have dropped the infinite tail of 9s and simply replaced them with the > so I have only perfromed operations on digits in finite postions, I have contained the infinte tail to the simple fact it is greater then 0.

Similarily I develope the fact that .999... - .1^n < 1

From these 2 inequalities I can make the statement

Absolute Value (1- .999....) < .1^n


What does this mean?

Simple... we can confine .999... to an arbitrally small distance from 1. There are no restrictions on n, this statement is true for ALL n where n is a positive integer.

The next question that must be answered is what do we mean by equality in terms of Real Analysis. It is not as easy when you are dealing with numbers such as the one we have here.

In short to show equality of z an y, where x and y are Real numbers, it is only necessary to show that y exists in every interval surrounding x. That is Absolute Value (x -y) <e for all e>0.

I think it is this definition of equality that is the stumpling block for this argument. While myself and other mathematicians on the board are familiar with and comfortable with this definition the non mathematicains are not and in fact do not have a working definition of equality. Thus we are simply not talking the same language.

Time for a beer!

 

[silverpig]

What's odder is your insistance that 1 is an objective concept. Furthermore, we're not dealing with objective vs. subjective here. All numbers are subjective. We made them up. People defined mathematics and all rules that go along with it. Without people, math does not exist. Infinity, 0, 1, 0.999..., pi, they're ALL subjective.

 

[RossGr]

I am not assigning a fixed value to infinity, but you seem to want to.

my definition of infinity is

infinity > x for all x in the Real numbers. How is that a fixed number and when have I ever used it as if it were?

Once again a strawman argument. I know that you will never adress my questions. We have been at this dissussion for nearly 2 years I believe. Counting, of course, the thread in HT.

 

[bleeb]

1234th POST!

 

[bleeb]

Lets keep this thing going!

 

[prometheusxls]

Originally posted by: Electrode
The flaw:

x = 0.9999...
10x = 9.9999...
10x - x = 9.9999... - 0.9999...
9x = 9
x = 1.

After the bold statement, x is considered to be the result of the bold statement, as opposed to 0.9999... as previously assumed, therefore the final statement is 1 = 1.

Yep you can't redfine the variable in the middle of the arguement and then refer to the original deffinition in conclusion.

 

[prometheusxls]

Originally posted by: Lonyo
If 10x - x gives us 9, then 9 = 9x.
If we multiply x by 9, does it equal 9? Only if x is 1 exactly, otherwise 9x doesn't equal 9.
9 x 0.99.... = 9? I don't see how really. Surely it would be 8.9999.....
The only way I see of getting 9 useing 9 is by 9 x 1

Exactly. Its a lousy argument.

 

[silverpig]

Originally posted by: prometheusxls

Originally posted by: Electrode
The flaw:

x = 0.9999...
10x = 9.9999...
10x - x = 9.9999... - 0.9999...
9x = 9
x = 1.

After the bold statement, x is considered to be the result of the bold statement, as opposed to 0.9999... as previously assumed, therefore the final statement is 1 = 1.

Yep you can't redfine the variable in the middle of the arguement and then refer to the original deffinition in conclusion.

Huh?

a = b
b = c

then a = c

Got a problem with that?

 

[silverpig]

Originally posted by: prometheusxls

Originally posted by: Lonyo
If 10x - x gives us 9, then 9 = 9x.
If we multiply x by 9, does it equal 9? Only if x is 1 exactly, otherwise 9x doesn't equal 9.
9 x 0.99.... = 9? I don't see how really. Surely it would be 8.9999.....
The only way I see of getting 9 useing 9 is by 9 x 1

Exactly. Its a lousy argument.

It's a perfectly fine argument. And yes, if you do the arithmetic out then 9 x 0.999... = 8.999... = 9 = 9 x 1

Nothing wrong with that either.

 

[dighn]

Originally posted by: prometheusxls

Originally posted by: Lonyo
If 10x - x gives us 9, then 9 = 9x.
If we multiply x by 9, does it equal 9? Only if x is 1 exactly, otherwise 9x doesn't equal 9.
9 x 0.99.... = 9? I don't see how really. Surely it would be 8.9999.....
The only way I see of getting 9 useing 9 is by 9 x 1

Exactly. Its a lousy argument.

granted that's no real proof butr i dont understand why people are having so much problems with that

but whatever. how abotu a version of that without the x eh?


9*0.9r = (10-1)*0.9r = 9.9r - 0.9r = 9 = 9*1
9*0.9r = 9*1
0.9r = 1

 

[silverpig]

Nice job there too dighn

 

[dighn]

thanks

i wonder when this thread will finally die. it's the same arguments over and over

 

[silverpig]

It will die when the AT trend of posting in threads before you've read them ends

 

[PHiuR]

geek forum ... oh wait this is hehehe

 

[MadRat]

9*0.9r = (10-1)*0.9r = 9.9r - 0.9r = 9 = 9*1
9*0.9r = 9*1
0.9r = 1

For one, the idea that .9r can be divisible by 10 is laughable. What happens is the final decimal place of the last nine is no longer at infinity, but actually it would become "(infinity-1)" by your argument. The "mathematicians" of the forum seem to conveniently ignore the idea of a limit to the 9's in .9r at the infinite position.

You also cannot factor out the 9 from .9r and presume to remain accurate to what you have defined in your original terms. The argument becomes flawed at this point.

 

[silverpig]

Originally posted by: MadRat

9*0.9r = (10-1)*0.9r = 9.9r - 0.9r = 9 = 9*1
9*0.9r = 9*1
0.9r = 1

For one, the idea that .9r can be divisible by 10 is laughable. What happens is the final decimal place of the last nine is no longer at infinity, but actually it would become "(infinity-1)" by your argument. The "mathematicians" of the forum seem to conveniently ignore the idea of a limit to the 9's in .9r at the infinite position.

You also cannot factor out the 9 from .9r and presume to remain accurate to what you have defined in your original terms. The argument becomes flawed at this point.

Infinity - 1 huh? Infinite position?

There is no infinitieth position. There is no infinity-minus-one-th position.

Are you trying to say that if you take 0.9r x 10 you'll get 0.9r...0 or something?

 

[shady123]

its not 1, no q

 

[MadRat]

Originally posted by: silverpig
Infinity - 1 huh? Infinite position?

There is no infinitieth position. There is no infinity-minus-one-th position.

Are you trying to say that if you take 0.9r x 10 you'll get 0.9r...0 or something?

Its very simple, the 9's extend to infinity, making the "last" nine at the position of "infinity". Infinity doesn't exist in a fixed position on the number line, which seems to confuse the math gurus. But that does not mean the term "infinite position" can not be used to describe the limit of the number. If the nines extended beyond infinity then the number would be equal to or greater than "one". Likewise, if the number of nines is shifted 'one less than infinity' then a measurable gap would theoretically be created between "one" and the number. Luckily for all of us, infinity is by definition, undefined. The division and multiplication of a repeating decimal is nonsensical in the sense that the position of the last number cannot be moved because its position is already undefined.